I thought the grid is the surface of where everything takes place and stands on.
The grid is not a floor or ground. It is not even an object. It is just a guide so that users can see the units used in the scene and to have a visual representation of the horizontal coordinates.
All objects have what is called an origin (a pivot point or a "center", usually represented by an orange dot) The origin of the the created object is then placed where the 3D cursor is on the scene. By default the cursor is at coordinate 0, 0, 0.
In other words, by default the object's center is placed at the center of the 3D space.
To have your objects appear in a different place, just move the 3D cursor.
The center of your object is added at the cursor's position. If you haven't changed it, it's set to (0, 0, 0). So parts of your object are above 0, some are below (for the z axis, but it's the same with x and y).
And the center (called origin) is by default set to the geometry center, not what you would probably call the "bottom" of your object.